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I have no source for this, but I was once taught the following rule: inequalities mix with Big-O notation with an implied for all for each Big-O term on the left, and an implied there exists for each Big-O term on the right. Let me explain myself with some examples.

Your inequality ($f(m) \le m^{O(1)}$) translates to "There exists a function $g(m) \in O(1)$ such that for all $m$, we have $f(m) \le m^{g(m)}$."

The inequality "$O(n) < O(n^2)$" translates to "For all functions $g(n) \in O(n)$, there exists a function $h(n) \in O(n^2)$ such that for all $n$, we have $g(n) < h(n)$."